A Transformation Based Algorithm for Reversible Logic Synthesis

1. A Transformation Based Algorithm for Reversible Logic Synthesis D. Michael Miller Dmitri Maslov Gerhard W. Dueck Design Automation Conference, 2003

2. Objectives • Synthesis of Toffoli cascades • reversible logic • application in • quantum computing • low power CMOS • optical computing • Approach • use a naive algorithm to find a solution • apply transformations to simplify the network

3. Toffoli Gate Family We use such gates for any number of inputs

4. The Basic Algorithm • Reversible function is given as a truth table • For each row in the truth table • Introduce Toffoli gates such that the output equals the input • Make sure that previous outputs are not affected

5. Observe that after applying S3 wires are as in input The Basic Algorithm c b a in 000 001 010 011 100 101 110 111 out 001 000 011 010 101 111 100 110 S1 000 001 010 011 100 110 101 111 S1 000 001 010 011 100 110 101 111 S1 000 001 010 011 100 110 101 111 S2 000 001 010 011 100 111 101 110 S3 000 001 010 011 100 101 111 110 S3 000 001 010 011 100 101 111 110 S3 000 001 010 011 100 101 110 111 Final circuit Read quantum array in inverse order to applying gates

6. Bidirectional Algorithm in 000 001 010 011 100 101 110 111 out 111 000 001 010 011 100 101 110 need three gates need only one gate

7. Bidirectional Algorithm c b a inputs S2 000 001 010 111 100 101 110 011 S3 000 001 010 011 100 101 110 111 S1 000 111 010 001 100 011 110 101 in 000 001 010 011 100 101 110 111 out 111 000 001 010 011 100 101 110 ab Final circuit S1 After applying S3 my wires are the same as inputs

8. Improvements • Output permutations • via swap gates • Control input reduction • there may be more than on possible assignment of control variables • select the one that makes the function “simple” • Template matching

9. Templates • Idea: replace a sequence of gates with an equivalent shorter sequence • example:

10. Example

11. Template Classification

12. Larger Example rd53

13. Number of reversible functions using aspecified number of gates naive gates 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 Average 4 72 477 1759 4179 6912 8389 7766 5615 3183 1391 453 104 15 1 7.65 5 87 550 1901 4267 6828 8221 7670 5610 3204 1402 455 104 15 1 7.67 1 14 92 380 1113 2468 4311 6083 7044 6754 5379 3549 1922 839 286 72 12 1 8.67 permutation 3 86 493 2312 6944 11206 10169 5945 2375 650 121 15 1 6.53 5 110 792 4726 11199 12076 7518 2981 767 130 15 1 6.18 12 6817 17531 11194 3752 844 134 15 1 5.63 input control bidirectional templates optimal

14. Future Work • Start from a non-reversible function • Verify the completeness of the templates • Expand the algorithm to include Fredkin gates • Select the best permutation (not by exhaustive search)